Matematisk modell för optimal kontroll av läkemedel i ett tumör-immunsystem

Abstract

In this thesis, we consider a mathematical model, in the form of a system of ordinary differential equations (ODEs), which describes the dynamics of a tumor-immune system with chemotherapeutic as well as immunotherapeutic drugs. We develop an optimization method for the solution of a parameter identification problem (PIP) for the system of ODEs, in order to minimize the number of tumor cells as well as the chemotherapeutic and immunotherapeutic drug administration. We formulate the minimization problem for the Tikhonov regularization functional and for an objective functional, which was considered by biomedical researchers in previous publications. To minimize both functionals we use the Lagrangian approach and formulate the optimization algorithm, which involves solution of the adjoint and forward problems. The adjoint and forward problems are solved by using Newtons method, whose accuracy is implemented and tested. The methods for the solution of the problem are also formulated, verified and numerically tested in Matlab. The results give rise to a discussion that verify the accuracy of Newtons method and present it as useful for future work. Additionaly, the results show a need to improve the mathematical model in future work since is shown to be deficient. Furthermore, differences and similarities between the two functionals are discussed to determine which functional is preferable to use in future studies. Finally, epidemiological findings are anaylzed based on our analytical conclusions.